show that the axis are to be rotated through an angle of 1/2tan^-1(2h/a-b) so as to remove the XY term from the equation ax^2+2hxy+by^2=0. if a not equals to b and through the angle π/4 of a=b.
Answers
Answer:
Let the x and y axis be rotated by an angle is as shown below:
then, x=xcosθ−ysinθ
y=xsinθ+ycosθ, where (x, y) is the coordinates with respect to new coordinate axes.
Given: ax
2
+2hxy+by
2
+2gx+2fy+c=0
Replace x→xcosθ−ysinθ
y→xsinθ+ycosθ
⇒a(xcosθ−ysinθ)
2
+2h(xcosθ−ysinθ)(xsinθ+ycosθ)+b(xsinθ+ycosθ)
2
+2g(xcosθ−ysinθ)+2f(xsinθ+ycosθ)+c=0
⇒a(x
2
cos
2
θ−y
2
sin
2
θ)+2h(x
2
sinθcosθ+xycos
2
θ−xysin
2
θ−y
2
sinθcosθ)+b(x
2
sin
2
θ+y
2
cos
2
θ)+2g(xcosθ−ysinθ)+2f(xsinθ+ycosθ)+c=0
Now taking out every xy term
−2axysinθcosθ+2hxycos
2
θ−2hxysin
2
θ+2hxysinθcosθ
To eliminate the xy term, put coefficient of xy=0
⇒−2asinθcosθ+(2hcos
2
θ−2hsin
2
θ)+2hsinθcosθ=0
⇒2h(cos2θ)+sin2θ(b−a)=0
⇒tan2θ=
a−b
2h
⇒θ=
2
1
tan
−1
(
a−b
2h
).
Step-by-step explanation:
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